Question

Cold water enters a steam generator at \(20^{\circ} \mathrm{C}\) and leaves as saturated vapor at \(200^{\circ} \mathrm{C}\). Determine the fraction of heat used in the steam generator to preheat the liquid water from \(20^{\circ} \mathrm{C}\) to the saturation temperature of \(200^{\circ} \mathrm{C}\).

Short answer

Answer: Approximately 28.2% of the heat used in the steam generator is used to preheat the liquid water from 20°C to the saturation temperature of 200°C.

Step-by-step solution

Calculate the total heat input required to convert the water into saturated vapor

Total heat input (Qtotal) can be calculated using the formula: Qtotal = m × (Cp × ΔT + H_v) where m is the mass of water, Cp is the specific heat of water, ΔT is the temperature change, and H_v is the enthalpy of vaporization of water. We are not given the mass of water, but since we are looking for the fraction of heat used, we can assume any value for m as it will eventually cancel out in the calculations. Let's take the standard values of specific heat of water (Cp) as 4.18 kJ/kg·K and enthalpy of vaporization (H_v) as 2257 kJ/kg. ΔT = temperature difference = 200 - 20 = 180°C Now plugging the values into the formula, Qtotal = m × (4.18 kJ/kg·K ×180°C + 2257 kJ/kg)

Calculate the heat required to preheat the liquid water

Heat input for preheating (Qpreheat) can be calculated using the formula: Qpreheat = m × Cp × ΔT_preheat where ΔT_preheat is the temperature change during preheating. In this case, ΔT_preheat = 200 - 20 = 180°C Now plugging the values into the formula, Qpreheat = m × 4.18 kJ/kg·K × 180°C

Find the fraction of heat used for preheating

To find the fraction of heat used for preheating, we can divide the heat used for preheating by the total heat input: Fraction = Qpreheat / Qtotal Notice that the mass of water (m) will cancel out in the fraction: Fraction = (4.18 kJ/kg·K × 180°C) / (4.18 kJ/kg·K ×180°C + 2257 kJ/kg) Calculating the fraction, we get: Fraction ≈ 0.282 So, approximately 28.2% of the heat used in the steam generator is used to preheat the liquid water from 20°C to the saturation temperature of 200°C.

Key concepts

Specific Heat Capacity

When we describe how a substance reacts to the addition of heat, we often refer to its specific heat capacity, or simply specific heat. This value is a measure of how much heat energy is needed to raise the temperature of one kilogram of the substance by one degree Celsius (or Kelvin). Its units are typically expressed in joules per kilogram per degree Celsius (J/kg°C).

In our exercise, we're dealing with water, which has a high specific heat capacity of about 4.18 kJ/kg°C. This means that water needs a considerable amount of heat to increase in temperature, which is why it's an effective substance for transferring heat in systems like steam generators. The high specific heat capacity of water also means that, as it heats up, it can store a lot of energy, which is then released when the water cools down — playing a pivotal role in heating and cooling processes.

Enthalpy of Vaporization

The enthalpy of vaporization, often denoted as H_v, is the amount of heat that needs to be added to a liquid substance to transform it into a gas at constant pressure. In the context of water converting to steam in our exercise, it is a measure of the energy needed to turn liquid water at its boiling point (saturation temperature) to steam without changing the temperature.

This latent heat of vaporization is crucial in the steam generator, as it is the energy absorbed by water when it undergoes the phase change from liquid to vapor. For water, H_v is quite substantial - around 2257 kJ/kg, indicating a substantial amount of energy is necessary for the phase change. It's important to note that this value can vary depending on the pressure of the environment.

Saturation Temperature

Saturation temperature is the temperature at which a liquid boils and turns into vapor at a given pressure. It is one of the most important concepts in thermodynamics when discussing phase changes. The saturation temperature of water increases with pressure, which means at higher pressures, water will boil at temperatures above 100°C.

In our steam generator, the water exits as saturated vapor at 200°C. This indicates that the steam generator is operating under pressure conditions where the saturation temperature is elevated to 200°C, higher than water's standard boiling point at atmospheric pressure. It's essential for calculating the heat input required to cause the phase change from liquid water to steam within the generator.